We show that for a wide class of manifold pairs N,M with dim(M) = dim(N) + 1, every π1–injective map f : N →M factorises up to homotopy as a finite cover of an embedding. This result, in the spirit of Waldhausen’s torus theorem, is derived using Cappell’s surgery methods from a new algebraic splitting theorem for Poincaré duality groups. As an application we derive a new obstruction to the existence of π1–injective maps.